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\begin{document}

\title{Complex Analysis}
\subtitle{Chapter 4. Complex Integration}
%\institute{SLUC}
\author{LVA}
%\date
%\renewcommand{\today}{\number\year \,年 \number\month \,月 \number\day \,日}
%\date{ {2023年9月21日} }

\maketitle

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\begin{frame}{Contents 1-2}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{enumerate}

\item {\color{red}Fundamental Theorems}
\begin{enumerate}
\item[1.1.] Line Integrals
\item[1.2.] Rectifiable Arcs
\item[1.3.] Line Integrals as Functions of Arcs
\item[1.4.] Cauchy's Theorem for a Rectangle
\item[1.5.] Cauchy's Theorem in a Disk
\end{enumerate}
 
\item {\color{red}Cauchy's Integral Formula}
\begin{enumerate}
\item[2.1.] The Index of a Point with Respect to a Closed Curve
\item[2.2.] The Integral Formula
\item[2.3.] Higher Derivatives
\end{enumerate}

\end{enumerate}

\end{frame}

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\begin{frame}{Contents 3-4}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{enumerate}

\item[3.] {\color{red}Local Properties of Analytical Functions}
\begin{enumerate}
\item[3.1.] Removable Singularities. Taylor's Theorem
\item[3.2.] Zeros and Poles
\item[3.3.] The Local Mapping
\item[3.4.] The Maximum Principle
\end{enumerate}

\item[4.] {\color{red}The General Form of Cauchy's Theorem}
\begin{enumerate}
\item[4.1.] Chains and Cycles
\item[4.2.] Simple Connectivity
\item[4.3.] Homology
\item[4.4.] The General Statement of Cauchy's Theorem
\item[4.5.] Proof of Cauchy's Theorem
\item[4.6.] Locally Exact Differentials
\item[4.7.] Multiply Connected Regions
\end{enumerate}

\end{enumerate}

\end{frame}

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\begin{frame}{Contents 5-6}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{enumerate}

\item[5.] {\color{red}The Calculus of Residues}
\begin{enumerate}
\item[5.1.] The Residue Theorem
\item[5.2.] The Argument Principle
\item[5.3.] Evaluation of Definite Integrals
\end{enumerate}

\item[6.] {\color{red}Harmonic Functions}
\begin{enumerate}
\item[6.1.] Definition and Basic Properties
\item[6.2.] The Mean-value Property
\item[6.3.] Poisson's Formula
\item[6.4.] Schwarz's Theorem
\item[6.5.] The Reflection Principle
\end{enumerate}

\end{enumerate}

\end{frame}


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\begin{frame}{6.1. Definition and Basic Properties.  }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
What is a harmonic function?
}

\item  Answer. 
\begin{enumerate}
\item 
A real-valued function $u(z)$ or $u(x,y)$, defined and single-valued in a region $\Omega$, is said to be harmonic in $\Omega$, or a potential function, if it is continuous together with its partial derivatives of the first two orders and satisfies Laplace's equation
$$
\Delta u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0.
$$

\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{6.1. Definition and Basic Properties. Theorem 19. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
If $u_1$ and $u_2$ are harmonic in a region $\Omega$, then
%(60)
$$
\int_\gamma u_1^* du_2 - u_2^*du_1 = 0
$$
for every cycle $\gamma$ which is homologous to zero in $\Omega$.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{6.2. The Mean-value Property. Theorem 20. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
The arithmetic mean of a harmonic function over concentric circles $|z| = r$ is a linear function of $\log r$,
%(61)
$$
\frac{1}{2\pi} \int_{|z|=r} ud\theta = \alpha \log r + \beta, 
$$
and if $u$ is harmonic in a disk $\alpha = 0$ and the arithmetic mean is constant.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{6.2. The Mean-value Property. Theorem 21. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
A nonconstant harmonic function has neither a maximum nor a minimum in its region of definition. Consequently, the maximum and the minimum on a closed bounded set $E$ are taken on the boundary of $E$.

}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{6.2. The Mean-value Property. Exercise - 1 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
If $u$ is harmonic and bounded in $0 < |zl < \rho$, show that the origin is a removable singularity in the sense that $u$ becomes harmonic in $|z| < \rho$ 
when $u(0)$ is properly defined.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{6.2. The Mean-value Property. Exercise - 2 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
Suppose that $f(z)$ is analytic in the annulus $r_1 < |z| < r_2$ and continuous on the closed annulus. If $M(r)$ denotes the maximum of $|f(z)|$ for $|z| = r$,  show that $$M(r) \le M(r_1)^\alpha M(r_2)^{1-\alpha}$$ 
where $\alpha = \log (r_2/r): \log (r_2/r_1)$.
Discuss cases of equality. 
}

\item  Answer. 
\begin{enumerate}
\item Hadamard's three-circle theorem. 
 
\item Apply the maximum principle to a linear combination of $\log |f(z)|$ and $\log |z|$. 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{6.3. Poisson's Formula. Theorem 22. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
Suppose that $u(z)$ is harmonic for $|z| < R$, continuous for $|z|\le R$. 
Then
%(64)
$$
u(a) = \frac{1}{2\pi} \int_{|z|=R} \frac{R^2-|a|^2}{|z-a|^2}u(z)d\theta 
$$
for all $|a| < R$.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{6.4. Schwarz's Theorem. Theorem 23. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
For any piecewize continuous function $U(\theta)$ in $0\le \theta\le 2\pi$, the function 
$$
P_U(z) = \frac{1}{2\pi} \int_0^{2\pi} 
\mathrm{Re}\frac{e^{i\theta}+z}{e^{i\theta}-z} U(\theta)d\theta
$$ 
is harmonic for $|z| < 1$, and
%(68)
$$
\lim\limits_{z\to e^{i\theta_0}} P_U(z) = U(\theta_0)
$$
provided that $U$ is continuous at $\theta_0$.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{6.4. Schwarz's Theorem.  }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
There is an interesting geometric interpretation of Poisson's formula, also due to Schwarz.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{6.4. Schwarz's Theorem. Exercise - 1 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
Assume that $U(\xi)$ is piecewise continuous and bounded for all real $\xi$. Show that
$$P_U(z) = \frac{1}{\pi} \int_{-\infty}^{\infty} \frac{y}{(x-\xi)^2+y^2}U(\xi)d\xi$$
represents a harmonic function in the upper half plane with boundary
values $U(\xi)$ at points of continuity (Poisson's integral for the half plane).
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{6.4. Schwarz's Theorem. Exercise - 2 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
Prove that a function which is harmonic and bounded in the upper
half plane, continuous on the real axis, can be represented as a Poisson
integral (Ex. 1).
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{6.4. Schwarz's Theorem. Exercise - 3 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
In Ex. 1, assume that $U$ has a jump at $0$, for instance $U(+0) = 0$, $U(-0) = 1$.
Show that $P_u(z) -\frac{1}{\pi} \mathrm{arg} z$ tends to 0 as $z\to 0$.
Generalize to arbitrary jumps and to the case of the circle.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{6.4. Schwarz's Theorem. Exercise - 4 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
If $C_1$ and $C_2$ are complementary arcs on the unit circle, set $U = 1$ 
on $C_1$, $U = 0$ on $C_2$. Find $P_u(z)$ explicitly and show that $2\pi P_u(z)$ equals
the length of the arc, opposite to $C_1$, cut off by the straight lines through $z$ and the end points of $C_1$.

}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{6.4. Schwarz's Theorem. Exercise - 5 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
Show that the mean-value formula (62) remains valid for $u = \log|1+z|$, $z_0=0$, $r=1$, and use this fact to compute $$ \int_0^\pi \log \sin\theta d\theta. $$
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{6.4. Schwarz's Theorem. Exercise - 6 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
If $f(z)$ is analytic in the whole plane and if $z^{-1} \mathrm{Re} f(z)\to 0$ when
$z\to\infty$, show that $f$ is a constant. Hint: Use (66).
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{6.4. Schwarz's Theorem. Exercise - 7 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
If $f(z)$ is analytic in a neighborhood of $\infty$ and if $z^{-1}\mathrm{Re} f(z)\to 0$ when $z\to\infty$, show that $\lim\limits_{z\to\infty} f(z)$ exists. 
(In other words, the isolated singularity at $\infty$ is removable.)
Hint: Show first, by use of Cauchy's integral formula, that $f = f_1 + f_2$ 
where $f_1(z)\to 0$ for $z\to\infty$ and $f_2(z)$ is analytic in the whole plane.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{6.4. Schwarz's Theorem. Exercise - 8 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
If $u(z)$ is harmonic for $0 < |z| < \rho$ and $\lim\limits_{z\to 0} zu(z) = 0$, prove that $u$ can be written in the form $u(z) = a \log |z| + u_0(z)$ where $\alpha$ is a constant and $u_0$ is harmonic in $|z| < \rho$.

Hint: Choose $\alpha$ as in (61). Then show that $u_0$ is the real part of an analytic function $f_0(z)$ and use the preceding exercise to conclude that the singularity at $0$ is removable.

}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{6.5. The Reflection Principle. Theorem 24. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
Let $\Omega^+$ be the part in the upper half plane of a symmetric region $\Omega$, and let $\sigma$ be the part of the real axis in $\Omega$. 
Suppose that $v(x)$ is continuous in $\Omega^+\cup \sigma$, harmonic in $\Omega^+$, and zero on $\sigma$. 
Then $v$ has a harmonic extension to $\Omega$ which satisfies the symmetry relation $v(\bar{z}) = -v(z)$. 
In the same situation, if $v$ is the imaginary part of an analytic function $f(z)$ in $\Omega^+$, then $f(z)$ has an analytic extension which satisfies $f(z) = \overline{f(\bar{z})}$.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{6.5. The Reflection Principle. Exercise - 1 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
If $f(z)$ is analytic in the whole plane and real on the real axis, purely imaginary on the imaginary axis, show that $f(z)$ is odd.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{6.5. The Reflection Principle. Exercise - 2 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
Show that every function $f$ which is analytic in a symmetric region 
$\Omega$ can be written in the form $f_1 + if_2$ where $f_1,f_2$ are analytic in $\Omega$ and real on the real axis.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{6.5. The Reflection Principle. Exercise - 3 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
If $f(z)$ is analytic in $|z|\le 1$ and satisfies $|f| = 1$ on $|z| = 1$, show that $f(z)$ is rational.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{6.5. The Reflection Principle. Exercise - 4 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
Use %(66)
$$
f(z) = \frac{1}{2\pi i} \int_{|\zeta|=R} \frac{\zeta+z}{\zeta-z}u(\zeta)\frac{d\zeta}{\zeta} + iC
$$
to derive a formula for $f'(z)$ in terms of $u(z)$.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{6.5. The Reflection Principle. Exercise - 5 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
If $u(z)$ is harmonic and $0\le u(z)\le Ky$ for $y > 0$, prove that $u = ky$ with $0\le k\le K$. 
}

\item  Answer. 
\begin{enumerate}
\item 
Reflect over the real axis, complete to an analytic function $f(z) = u + iv$, and use Ex. 4 to show that $f'(z)$ is bounded.
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\end{document}

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